Simplify and expand the following expression: $ \dfrac{z}{z - 3}-\dfrac{5z - 7}{z + 10} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(z - 3)(z + 10)$ Multiply the first term by $\dfrac{z + 10}{z + 10}$ $ \begin{align*} \dfrac{z}{z - 3} \times \dfrac{z + 10}{z + 10} & = \dfrac{(z)(z + 10)}{(z - 3)(z + 10)} \\ & = \dfrac{z^2 + 10z}{(z - 3)(z + 10)}\end{align*} $ Multiply the second term by $\dfrac{z - 3}{z - 3}$ $ \begin{align*} \dfrac{5z - 7}{z + 10} \times \dfrac{z - 3}{z - 3} & = \dfrac{(5z - 7)(z - 3)}{(z + 10)(z - 3)} \\ & = \dfrac{5z^2 - 22z + 21}{(z + 10)(z - 3)}\end{align*} $ Now we have: $ = \dfrac{z^2 + 10z}{(z - 3)(z + 10)} - \dfrac{5z^2 - 22z + 21}{(z + 10)(z - 3)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{z^2 + 10z - (5z^2 - 22z + 21)}{(z - 3)(z + 10)} $ $ = \dfrac{z^2 + 10z - 5z^2 + 22z - 21}{(z - 3)(z + 10)} $ $ = \dfrac{-4z^2 + 32z - 21}{(z - 3)(z + 10)}$ Expand the denominator: $ = \dfrac{-4z^2 + 32z - 21}{z^2 + 7z - 30}$